Abstract:
In the first part we survey basic results of Geometric Invariant Theory.
Classical Mumford's construction of the quotient
of the set of semistable points, variation of quotients, combinatorial
methods in the case of torus actions, and
classification results under certain restrictions on geometry of the
quotient space will be discussed.
The second part is devoted to a useful invariant of an algebraic variety
called the total coordinate ring
or the Cox ring. We consider such algebraic properties of Cox rings as
factoriality, graded factoriality,
and finite generation. A canonical quotient presentation of a variety
will be defined and applications
of this construction will be illustrated on concrete examples.
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